NOFX are good too, I love bands who don't take themselves seriously. Haven't heard much Green Day, but Basket Case is a classic (then again, they also have that song with the "I hope you had the time of your life" part, ugh).
NOFX is great. They haven't made a bad album in 25 years and they're one of the funniest and most entertaining live bands you'll ever see. Especially their albums 'Punk in Drublic' and 'So Long and Thans for All the Shoes' are great. Also be sure to check out their 'Backstage Passport' documentary. It's about a really crazy world tour they did a few years ago. NOFX figured that their tours were too safe and smooth, they wanted to do another tour that would be a complete disaster like in the old days. So they got a few really shady South American promotors with bad reputations to set up some shows for them, went to Asian countries where virtually nobody ever heard of NOFX, and to countries where they had a good chance of getting in trouble with authorities simply for being a punk band. And they brought a chronically drunk road manager. That's one of the greatest rock 'n' roll adventures you'll ever see.
'Kerplunk!' and 'Dookie' by Green Day are excellent, lots of really great songs on those albums. '39/Smooth' and 'Insomiac' have a few good songs too. But you can live without everything they did after that.